Simplify and expand the following expression: $ \dfrac{2k - 1}{5k - 3}+\dfrac{3k + 9}{4k + 3} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5k - 3)(4k + 3)$ Multiply the first term by $\dfrac{4k + 3}{4k + 3}$ $ \begin{align*} \dfrac{2k - 1}{5k - 3} \times \dfrac{4k + 3}{4k + 3} & = \dfrac{(2k - 1)(4k + 3)}{(5k - 3)(4k + 3)} \\ & = \dfrac{8k^2 + 2k - 3}{(5k - 3)(4k + 3)}\end{align*} $ Multiply the second term by $\dfrac{5k - 3}{5k - 3}$ $ \begin{align*} \dfrac{3k + 9}{4k + 3} \times \dfrac{5k - 3}{5k - 3} & = \dfrac{(3k + 9)(5k - 3)}{(4k + 3)(5k - 3)} \\ & = \dfrac{15k^2 + 36k - 27}{(4k + 3)(5k - 3)}\end{align*} $ Now we have: $ = \dfrac{8k^2 + 2k - 3}{(5k - 3)(4k + 3)} + \dfrac{15k^2 + 36k - 27}{(4k + 3)(5k - 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{8k^2 + 2k - 3 + 15k^2 + 36k - 27}{(5k - 3)(4k + 3)} $ $ = \dfrac{23k^2 + 38k - 30}{(5k - 3)(4k + 3)}$ Expand the denominator: $ = \dfrac{23k^2 + 38k - 30}{20k^2 + 3k - 9}$